Difficulty: Introductory | Prerequisites: None
Chemistry runs on precise measurement, and this first chunk of CHEM101 builds the toolkit: metric prefixes, scientific notation, significant figures, and dimensional analysis. Master these and every calculation in the course becomes a unit-conversion exercise. Get them wrong and your answers will be off by orders of magnitude.
Metric system
The standardised decimal system of measurement used in science. All units scale by powers of ten, which makes conversions straightforward once you know the prefixes.
In simple terms, this means: every unit is just a base unit (metre, gram, litre) multiplied or divided by 10, 100, 1000, and so on.
Scientific notation
A way of expressing very large or very small numbers as a coefficient (between 1 and 10) multiplied by a power of ten.
Think of it as shorthand: instead of writing 0.00056, you write 5.6 x 10^-4.
Significant figures (sig figs)
The number of meaningful digits in a measured or calculated quantity. They communicate how precise a measurement is.
In simple terms, this means: sig figs tell you which digits you can trust.
Precision
How close repeated measurements are to each other.
Think of it as consistency: if you weigh the same sample five times and get 2.31 g, 2.32 g, 2.31 g, 2.32 g, 2.31 g, your measurements are precise.
Accuracy
How close a measurement is to the true or accepted value.
Think of it as correctness: a precise set of measurements can still be inaccurate if the balance was poorly calibrated.
Density
The ratio of an object's mass to its volume (d = m/V), expressed in units such as g/mL or g/cm cubed.
In simple terms, this means: density tells you how much stuff is packed into a given space.
Dimensional analysis
A problem-solving method that uses conversion factors to cancel unwanted units and arrive at the desired unit.
Think of it as a unit-tracking assembly line: multiply by fractions equal to 1 (e.g. 1 kg / 1000 g) until only the unit you want remains.
The metric system scales base units (gram, metre, litre) with prefixes. You need to know these cold:
M (Mega) = 10^6 (one million)
k (kilo) = 10^3 (one thousand)
d (deci) = 10^-1 (one tenth)
c (centi) = 10^-2 (one hundredth)
m (milli) = 10^-3 (one thousandth)
mu (micro) = 10^-6 (one millionth)
n (nano) = 10^-9 (one billionth)
p (pico) = 10^-12 (one trillionth)
Moving between prefixes is a matter of shifting the decimal point. To go from milli to micro, multiply by 1000 (three decimal places right). To go from kilo to base, multiply by 1000.
Mass is measured with a balance or scale, reported in grams (g). This is a direct measurement, not a derived one.
Liquids: use a graduated cylinder or volumetric flask. Read the meniscus at eye level.
Irregularly shaped solids: use water displacement. Submerge the object in a known volume of water and measure the rise.
Density = mass / volume, typically in g/mL or g/cm cubed.
Water's density is 1 g/mL at room temperature. This is the reference point: substances denser than water sink, those less dense float.
Density is an intensive property, which means it does not change with the amount of substance. A teaspoon and a bathtub of pure water both have a density of 1 g/mL.
Scientific notation expresses a number as a coefficient between 1 and 10, multiplied by a power of ten.
0.00056 becomes 5.6 x 10^-4 (decimal moved four places right, exponent is -4)
45,000 becomes 4.5 x 10^4 (decimal moved four places left, exponent is +4)
Rules of exponents for scientific notation:
Multiplying: add the exponents. (2 x 10^3)(3 x 10^4) = 6 x 10^7
Dividing: subtract the exponents. (6 x 10^8) / (2 x 10^3) = 3 x 10^5
Raising to a power: multiply the exponent. (2 x 10^3)^2 = 4 x 10^6
Sig figs tell you how reliable a number is. The rules for counting them:
All non-zero digits are significant. (245 has 3 sig figs)
Zeros between non-zero digits are significant. (1002 has 4 sig figs)
Leading zeros are never significant. (0.0045 has 2 sig figs)
Trailing zeros after a decimal point are significant. (2.300 has 4 sig figs)
Trailing zeros in a whole number without a decimal point are ambiguous. (1500 could be 2, 3, or 4 sig figs; use scientific notation to be clear)
Multiplication and division: the answer has the same number of sig figs as the measurement with the fewest sig figs.
Addition and subtraction: the answer has the same number of decimal places as the measurement with the fewest decimal places.
Precision is about reproducibility (how close repeated measurements are to each other). Accuracy is about correctness (how close a measurement is to the true value).
You can be precise without being accurate, and accurate without being precise. The ideal is both.
Dimensional analysis is the single most useful problem-solving technique in general chemistry. You set up conversion factors (fractions equal to 1) so that unwanted units cancel and the desired unit remains.
5 mL = 1 teaspoon
1 kg = 2.2 lbs
1 inch = 2.54 cm (exact)
Within the metric system, move the decimal by the difference in prefix powers (e.g. 1 mL = 1000 microlitres, because milli is 10^-3 and micro is 10^-6, a factor of 10^3 apart)
Write down the given quantity with its unit.
Multiply by a conversion factor arranged so the given unit is in the denominator (it cancels).
Repeat with additional conversion factors until only the target unit remains.
Multiply across the top, divide across the bottom.
Example: convert 5.0 kg to pounds.
5.0 kg x (2.2 lbs / 1 kg) = 11 lbs
The kg cancels, leaving lbs. Two sig figs in the given quantity, so the answer is reported as 11 lbs (2 sig figs).
Density: d = m / V
Scientific notation form: a x 10^n, where 1 <= a < 10
Metric prefix ladder (largest to smallest): M (10^6) > k (10^3) > base (10^0) > d (10^-1) > c (10^-2) > m (10^-3) > mu (10^-6) > n (10^-9) > p (10^-12)
Metric-to-English conversions:
1 inch = 2.54 cm (exact)
1 kg = 2.2 lbs
5 mL = 1 teaspoon
1 L = 1.057 quarts
Dimensional analysis is how pharmacists calculate drug dosages, how engineers size piping, and how pilots convert between nautical and statute miles. The method never changes; only the conversion factors do.
Density is the reason oil floats on water and why a helium balloon rises. It is also how geologists identify unknown mineral samples in the field: measure the mass, measure the volume, divide, and compare to known densities.
Students often confuse precision with accuracy. A set of measurements clustered tightly together is precise, but if they are all far from the true value, they are not accurate. The two are independent.
Leading zeros are not significant. Students frequently count 0.0045 as four sig figs when it has only two.
Trailing zeros in whole numbers (e.g. 1500) are ambiguous without a decimal point or scientific notation. If your professor writes 1500., the decimal point signals four sig figs.
Students sometimes multiply or divide by a conversion factor upside down, which doubles the unit error rather than cancelling it. Always check that the unit you want to remove is in the opposite position (numerator vs. denominator).
⚠️ Sig fig rules for multiplication/division vs. addition/subtraction are different. Exams love mixing them in a single multi-step problem.
⚠️ Dimensional analysis problems are nearly guaranteed on every CHEM101 exam. Set up the conversion factor chain clearly and cancel units on paper.
⚠️ Know the density of water (1 g/mL at room temperature). It is used as a reference in many problems.
⚠️ Be able to convert between scientific notation and standard form in both directions, and know the exponent rules for multiplying, dividing, and raising to a power.
True or false: 0.00320 has four significant figures. (True: the 3, 2, and trailing 0 after the decimal are significant, plus the zero between 3 and 2.)
Fill in the blank: 1 km = ______ m. (1000)
True or false: precision and accuracy mean the same thing. (False)
Fill in the blank: density = ______ / ______. (mass / volume)
True or false: in scientific notation, the coefficient must be between 1 and 10. (True)
Q: A metal cube has a mass of 15.3 g and a volume of 2.0 mL. What is its density? Report with the correct number of significant figures.
A: d = m/V = 15.3 g / 2.0 mL = 7.65, rounded to 7.7 g/mL (2 sig figs, limited by the volume measurement).
Q: Convert 2500 microlitres to millilitres.
A: 2500 microlitres x (1 mL / 1000 microlitres) = 2.5 mL.
Q: Express 0.0000072 in scientific notation.
A: 7.2 x 10^-6.
Q: A student measures a sample's mass three times and gets 4.52 g, 4.53 g, and 4.51 g. The true mass is 4.80 g. Are these measurements precise, accurate, both, or neither?
A: Precise (they are clustered closely together) but not accurate (they are far from the true value of 4.80 g).
Q: How many significant figures are in 1.030 x 10^4?
A: Four. The trailing zero after 3 is significant because it follows a decimal point.
These measurement fundamentals feed directly into stoichiometry (later in CHEM101), where every calculation requires balancing units and tracking sig figs through multi-step problems.
Density links to the classification of matter and physical properties covered in the next set of notes. It is also the basis of many lab techniques, including the use of specific gravity in identifying unknown substances.
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